《復變數(第2版)》是Cambridge《套用數學系列叢書》之一，內容相當精闢，巧妙地展示了復變數在數學科學中的核心地位以及其在工程和物理科學套用中的關鍵性作用。復變數的引入不僅增加數學理論本身的完美性，更重要的是提供了一種解決一些數學疑難問題的途徑，甚至可以說是解決有些問題的唯一途徑。 《復變數(第2版)》的內容分為兩大部分。第一部分是整個課程的引入，包括：解析函式，積分，級數和殘數積分等初等理論以及一些過渡性方法：複平面的普通微分方程、數值方法等。第二部分包括保形映射，漸近擴張以及Riemann-Hilbert問題。每章節都提供了大量的套用、圖例以及練習，這些可以幫助讀者加深對復變數的基本概念和基本定理的理解。新版本做了全新的改進，是研究生以及分析方向本科生的理想教程。
基本介紹
- 書名：復變數
- 出版社：世界圖書出版公司
- 頁數：647頁
- 開本：24
- 定價：69.00
- 作者：M.J) 阿布婁韋提茲 (Ablowitz
- 出版日期：2008年3月1日
- 語種：英語
- ISBN：9787506291804
- 品牌：世界圖書出版公司北京公司
基本介紹,內容簡介,作者簡介,圖書目錄,
基本介紹
內容簡介
《復變數(第2版)》由世界圖書出版公司出版。
作者簡介
作者：(美國)阿布婁韋提茲(Ablowitz,M.J)
圖書目錄
Sections denoted with an asterisk (*) can be either omitted or read
independently.
Preface
PartⅠ Fundamentals and Techniques of Complex Function Theory
1 Complex Numbers and Elementary Functions
1.1 Complex Numbers and Their Properties
1.2 Elementary Functions and Stereographic Projections
1.2.1 Elementary Functions
1.2.2 Stereographic Projections
1.3 Limits, Continuity, and Complex Differentiation
1.4 Elementary Applications to Ordinary Differential Equations
2 Analytic Functions and Integration
2.1 Analytic Functions
2.1.1 The Cauchy-Riemann Equations
2.1.2 Ideal Fluid Flow
2.2 Multivalued Functions
*2.3 More Complicated Multivalued Functions and Riemann Surfaces
2.4 Complex Integration
2.5 Cauchy's Theorem
2.6 Cauchy's Integral Formula, Its a Generalization and Consequences
2.6.1 Cauchy's Integral Formula and Its Derivatives
*2.6.2 Liouville, Morera, and Maximum-Modulus Theorems
*2.6.3 Generalized Cauchy Formula and a Derivatives
*2.7 Theoretical Developments
3 Sequences, Series, and Singularities of Complex Functions
3.1 Definitions and Basic Properties of Complex Sequences,Series
3.2 Taylor Series
3.3 Laurent Series
*3.4 Theoretical Results for Sequences and Series
3.5 Singularities of Complex Functions
3.5.1 Analytic Continuation and Natural Barriers
*3.6 Infinite Products and Mittag-Leffler Expansions
*3.7 Differential Equations in the Complex Plane: Painleve Equations
*3.8 Computational Methods
*3.8.1 Laurent Series
*3.8.2 Differential Equations
4 Residue Calculus and Applications of Contour Integration
4.1 Cauchy Residue Theorem
4.2 Evaluation of Certain Definite Integrals
4.3 Principal Value Integrals and Integrals with Branch Points
4.3.1 Principal Value Integrals
4.3.2 Integrals with Branch Points
4.4 The Argument Principle, Rouche's Theorem
*4.5 Fourier and Laplace Transforms
*4.6 Applications of Transforms to Differential Equations
PartⅡ Applications of Complex Function Theory
5 Conformal Mappings and Applications
5.1 Introduction
5.2 Conformal Transformations
5.3 Critical Points and Inverse Mappings
5.4 Physical Applications
*5.5 Theoretical Considerations - Mapping Theorems
5.6 The Schwarz-Christoffel Transformation
5.7 Bilinear Transformations
*5.8 Mappings Involving Circular Arcs
5.9 Other Considerations
5.9.1 Rational Functions of the Second Degree
5.9.2 The Modulus of a Quadrilateral
*5.9.3 Computational Issues
6 Asymptotic Evaluation of Integrals
6.1 Introduction
6.1.1 Fundamental Concepts
6.1.2 Elementary Examples
6.2 Laplace Type Integrals
6.2.1 Integration by Parts
6.2.2 Watson's Lemma
6.2.3 Laplace's Method
6.3 Fourier Type Integrals
6.3.1 Integration by Parts
6.3.2 Analog of Watson's Lcmma
6.3.3 The Stationary Phase Method
6.4 The Method of Steepest Descent
6.4.1 Laplace's Method for Complex Contours
6.5 Applications
6.6 The Stokes Phenomenon
*6.6.1 Smoothing of Stokes Discontinuities
6.7 Related Techniques
*6.7.1 WKB Method
*6.7.2 The Mellin Transform Method
7 Riemann-Hiibert Problems
7.1 Introduction
7.2 Cauchy Type Integrals
7.3 Scalar Riemann-Hilbert Problems
7.3.1 Closed Contours
7.3.2 Open Contours
7.3.3 Singular Integral Equations
7.4 Applications of Scalar Riemann-Hilbert Problems
7.4.1 Riemann-Hilbert Problems on the Real Axis
7.4.2 The Fourier Transform
7.4.3 The Radon Transform
*7.5 Matrix Riemann-Hilbert Problems
7.5.1 The Riemann-Hilbert Problem for Rational Matrices
7.5.2 Inhomogeneous Riemann-Hilbert Problems and Singular Equations
7.5.3 The Riemann-Hilbert Problem for Triangular Matrices
7.5.4 Some Results on Zero Indices
7.6 The DBAR Problem
7.6.1 Generalized Analytic Functions
*7.7 Applications of Matrix Riemann-Hilbert Problems and Problems
Appendix A Answers to Odd-Numbered Exercises
Bibliography
Index
independently.
Preface
PartⅠ Fundamentals and Techniques of Complex Function Theory
1 Complex Numbers and Elementary Functions
1.1 Complex Numbers and Their Properties
1.2 Elementary Functions and Stereographic Projections
1.2.1 Elementary Functions
1.2.2 Stereographic Projections
1.3 Limits, Continuity, and Complex Differentiation
1.4 Elementary Applications to Ordinary Differential Equations
2 Analytic Functions and Integration
2.1 Analytic Functions
2.1.1 The Cauchy-Riemann Equations
2.1.2 Ideal Fluid Flow
2.2 Multivalued Functions
*2.3 More Complicated Multivalued Functions and Riemann Surfaces
2.4 Complex Integration
2.5 Cauchy's Theorem
2.6 Cauchy's Integral Formula, Its a Generalization and Consequences
2.6.1 Cauchy's Integral Formula and Its Derivatives
*2.6.2 Liouville, Morera, and Maximum-Modulus Theorems
*2.6.3 Generalized Cauchy Formula and a Derivatives
*2.7 Theoretical Developments
3 Sequences, Series, and Singularities of Complex Functions
3.1 Definitions and Basic Properties of Complex Sequences,Series
3.2 Taylor Series
3.3 Laurent Series
*3.4 Theoretical Results for Sequences and Series
3.5 Singularities of Complex Functions
3.5.1 Analytic Continuation and Natural Barriers
*3.6 Infinite Products and Mittag-Leffler Expansions
*3.7 Differential Equations in the Complex Plane: Painleve Equations
*3.8 Computational Methods
*3.8.1 Laurent Series
*3.8.2 Differential Equations
4 Residue Calculus and Applications of Contour Integration
4.1 Cauchy Residue Theorem
4.2 Evaluation of Certain Definite Integrals
4.3 Principal Value Integrals and Integrals with Branch Points
4.3.1 Principal Value Integrals
4.3.2 Integrals with Branch Points
4.4 The Argument Principle, Rouche's Theorem
*4.5 Fourier and Laplace Transforms
*4.6 Applications of Transforms to Differential Equations
PartⅡ Applications of Complex Function Theory
5 Conformal Mappings and Applications
5.1 Introduction
5.2 Conformal Transformations
5.3 Critical Points and Inverse Mappings
5.4 Physical Applications
*5.5 Theoretical Considerations - Mapping Theorems
5.6 The Schwarz-Christoffel Transformation
5.7 Bilinear Transformations
*5.8 Mappings Involving Circular Arcs
5.9 Other Considerations
5.9.1 Rational Functions of the Second Degree
5.9.2 The Modulus of a Quadrilateral
*5.9.3 Computational Issues
6 Asymptotic Evaluation of Integrals
6.1 Introduction
6.1.1 Fundamental Concepts
6.1.2 Elementary Examples
6.2 Laplace Type Integrals
6.2.1 Integration by Parts
6.2.2 Watson's Lemma
6.2.3 Laplace's Method
6.3 Fourier Type Integrals
6.3.1 Integration by Parts
6.3.2 Analog of Watson's Lcmma
6.3.3 The Stationary Phase Method
6.4 The Method of Steepest Descent
6.4.1 Laplace's Method for Complex Contours
6.5 Applications
6.6 The Stokes Phenomenon
*6.6.1 Smoothing of Stokes Discontinuities
6.7 Related Techniques
*6.7.1 WKB Method
*6.7.2 The Mellin Transform Method
7 Riemann-Hiibert Problems
7.1 Introduction
7.2 Cauchy Type Integrals
7.3 Scalar Riemann-Hilbert Problems
7.3.1 Closed Contours
7.3.2 Open Contours
7.3.3 Singular Integral Equations
7.4 Applications of Scalar Riemann-Hilbert Problems
7.4.1 Riemann-Hilbert Problems on the Real Axis
7.4.2 The Fourier Transform
7.4.3 The Radon Transform
*7.5 Matrix Riemann-Hilbert Problems
7.5.1 The Riemann-Hilbert Problem for Rational Matrices
7.5.2 Inhomogeneous Riemann-Hilbert Problems and Singular Equations
7.5.3 The Riemann-Hilbert Problem for Triangular Matrices
7.5.4 Some Results on Zero Indices
7.6 The DBAR Problem
7.6.1 Generalized Analytic Functions
*7.7 Applications of Matrix Riemann-Hilbert Problems and Problems
Appendix A Answers to Odd-Numbered Exercises
Bibliography
Index
